### A bornological approach to rotundity and smoothness applied to approximation.

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We state and prove a chain rule formula for the composition $T\left(u\right)$ of a vector-valued function $u\in {W}^{1,r}\left(\mathrm{\Omega};{\mathbb{R}}^{M}\right)$ by a globally Lipschitz-continuous, piecewise ${C}^{1}$ function $T$. We also prove that the map $u\to T\left(u\right)$ is continuous from ${W}^{1,r}\left(\mathrm{\Omega};{\mathbb{R}}^{M}\right)$ into ${W}^{1,r}\left(\mathrm{\Omega}\right)$ for the strong topologies of these spaces.

We introduce a criterion for a set to be Γ-null. Using it we give a shorter proof of the result that the set of points where a continuous convex function on a separable Asplund space is not Fréchet differentiable is Γ-null. Our criterion also implies a new result about Gâteaux (and Hadamard) differentiability of quasiconvex functions.

We construct a Lipschitz function f on X = ℝ ² such that, for each 0 ≠ v ∈ X, the function f is ${C}^{\infty}$ smooth on a.e. line parallel to v and f is Gâteaux non-differentiable at all points of X except a first category set. Consequently, the same holds if X (with dimX > 1) is an arbitrary Banach space and “a.e.” has any usual “measure sense”. This example gives an answer to a natural question concerning the author’s recent study of linearly essentially smooth functions (which generalize essentially smooth...

We give a relatively simple (self-contained) proof that every real-valued Lipschitz function on ${\ell}_{2}$ (or more generally on an Asplund space) has points of Fréchet differentiability. Somewhat more generally, we show that a real-valued Lipschitz function on a separable Banach space has points of Fréchet differentiability provided that the ${w}^{*}$ closure of the set of its points of Gâteaux differentiability is norm separable.

This paper deals with homeomorphisms F: X → Y, between Banach spaces X and Y, which are of the form $F\left(x\right):=F\u0303{x}^{(2n+1)}$ where $F\u0303:{X}^{2n+1}\to Y$ is a continuous (2n+1)-linear operator.

We show that every Lipschitz map defined on an open subset of the Banach space C(K), where K is a scattered compactum, with values in a Banach space with the Radon-Nikodym property, has a point of Fréchet differentiability. This is a strengthening of the result of Lindenstrauss and Preiss who proved that for countable compacta. As a consequence of the above and a result of Arvanitakis we prove that Lipschitz functions on certain function spaces are Gâteaux differentiable.

Let $f$ be a Lipschitz function on a superreflexive Banach space $X$. We prove that then the set of points of $X$ at which $f$ has no intermediate derivative is not only a first category set (which was proved by M. Fabian and D. Preiss for much more general spaces $X$), but it is even $\sigma $-porous in a rather strong sense. In fact, we prove the result even for a stronger notion of uniform intermediate derivative which was defined by J.R. Giles and S. Sciffer.

Modificando adecuadamente el método de un trabajo olvidado [1], probamos que si una aplicación continua, de un subconjunto abierto no vacío U de un espacio vectorial topológico metrizable separable y de Baire E, en un espacio localmente convexo, es direccionalmente diferenciable por la derecha en U según un subconjunto comagro de E, entonces, es genéricamente Gâteaux diferenciable en U. Nuestro resultado implica que cualquier espacio vectorial topológico, metrizable, separable y de Baire, es débilmente...

We give an example of a fourth degree polynomial which does not satisfy Rolle’s Theorem in the unit ball of ${l}_{2}$.